Nonlinear problems are interesting to engineers, biologists, physicists, mathematicians, and many other scientists. In nature, most systems are nonlinear. Nonlinear dynamical systems, describe changes in variables over time. Such systems may appear chaotic, unpredictable, or counterintuitive. Linear systems are much simpler.
A nonlinear system of equations is used to mathematically describe a nonlinear system. This is set of equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.
Nonlinear dynamical equations are difficult to solve. For this reason, they are commonly approximated using linear equations. If only limited accuracy or a range of the input values is required, this works well. The problem is that it hides interesting phenomena such as solitons, chaos, and singularities.
Although such chaotic behavior may appear random, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.
Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:
Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.
References[change | change source]
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